Mattress/box Spring Recommendation for Big Boy Bed from Toddler Bed

I'd decide what you want to spend, then go from there. If you have an Original Matress Factory near you, that's where I'd go.
I'd get a box spring/mattress set. As good of O. as you can afford.
The OMF bed is the best bed in our house.
I'm looking to replace my sons twin set with a full soon and I'll definitely go there for his.
I wouldn't get memory foam though. Just me.
Good luck!

ETA: they might make split box springs for full, like they do for queen/kings.

I've never used memory foam, but I would not worry about using a standard mattress and box springs. That's what our kids have, and they are 4 and 7.

We bought our mattresses and Denver Mattress. They are one of the cheapest I know of. I have to say, if I had it to do over again I probably would have chosen to pay more for the bed in the master bedroom. The two of us have caused it to sink in where we sleep, so there's this bump in the middle! But the kids' beds are just fine. Kids are little and don't need near the support adults do.

Our kids went straight from the crib to a regular bed, so I really think you don't need to worry too much about height. He'll be just fine!

There is memory foam and memory foam. Be sure to find out what you're buying. If they are inexpensive I suggest that they are not truly memory foam. A memory foam mattress is very heavy. Again, if they're light weight they're likely not memory foam.

If you have slats layed perpendicular to the side pieces of the frame then you don't need box springs. My grandchildren have a loft bed and a bunk bed set and they have the slats about 3 inches apart and use only a mattress. They have foam mattresses which are different than memory foam. They've held up well, lasting 8 years of so. One, we bought from Ikea.

The other mattress is called a bunk bed mattress. It's not as thick. It's firm and very light weight and flexible. It's solid foam and was used when my daughter, their mother, was a child. Kids are so light weight that mattresses last longer. This mattress was around $50 many years ago. I've seen similar ones a couple of years ago for less than $100.

A true me,pry foam mattress is expensive and weighs a ton. When I bought my goys mattresses, I opted for the best and extra long knowing they would be extremely tall. Best thing I ever did. They were 6'5". These
Lasted until they moved out. You can get a cheap foam one but it will not last, so it really depends on budget and how often you want to replace.

It sounds like you didn't get all the parts or something.

The box spring is hard so it supports the soft mattress. They are usually required unless the mattress is sitting on a piece of wood or a flat strong surface.